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© 2011 Carnegie Learning © 2011 Carnegie Learning

4.5 Developing the Graph of a Piecewise Function 255

Piecewise Functions

Developing the Graph of a Piecewise Function

Key Term

piecewise function

Learning Goals

In this lesson, you will:

Develop the graph of a piecewise function from a context with or without a table of values.

Represent a piecewise function algebraically by using appropriate notation, equations, and their domains.

Graph piecewise functions from contexts with or without diagrams.

Physically model the graphs of piecewise functions using technology.

T

he fairy tale Hansel and Gretel tells the story of a brother and sister who become lost in the woods and find a house made of candy. As the two children begin eating the candy house, a witch, who lives in the house, captures them and keeps them hostage in order to eat them.

As the story goes, the children left a trail of breadcrumbs from their home and into the woods so that they could find their way back. What similarities and differences are there between this trail of breadcrumbs and the kinds of graphs you have been creating? Create a graph showing a trail of breadcrumbs. How will you label the axes?

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© 2011 Carnegie Learning

Problem 1

Students generate a table, equation, and graph from a linear context. They will analyze all aspects of the context, including the intercepts, slope, domain, and range of the function, and domain and range of the problem situation.

Grouping

Have students complete Questions 1 through 6 with a partner. Then share the responses as a class.

Share Phase, Questions 1 and 2

How did you determine how much money you will have left?

What operation did you use to determine how much money you will have left?

As the x-values increase, what happens to the y-values?

As the y-values decrease, what happens to the x-values?

How can you tell from the table that the graph will be a line?

256 Chapter 4 Multiple Representations of Linear Functions

© 2011 Carnegie Learning

Problem 1

Investigating a Line

1. Suppose you earn $48 helping your neighbor with yard work. You think that you will spend $3 of your earnings each day.

a. How much money will you have left after 3 days? Show your work.

48 2 3(3) 5 48 2 9 5 39 I will have $39 left.

b. How much money will you have left after 5 days? Show your work.

48 2 3(5) 5 48 2 15 5 33 I will have $33 left.

c. How much money will you have left after 10 days? Show your work.

48 2 3(10) 5 48 2 30 5 18 I will have $18 left.

2. Complete the table to show the amounts of money you will have left for different numbers of days.

Time since You Started Spending (days)

Amount of Money Left (dollars)

0 48

1 45

2 42

3 39

4 36

5 33

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© 2011 Carnegie Learning

Share Phase,

Questions 3 through 6

What information in the problem situation did you use to determine the upper and lower bounds?

Is the scale you chose compatible with using rise ____ run to graph your line? Explain.

The directions said to “create a graph of your equation”.

Would the graph look any differently if the directions said to “create a graph representing the context”?

© 2011 Carnegie Learning

4.5 Developing the Graph of a Piecewise Function 257 3. Write an equation that represents the amount of money you have left in terms of the

number of days since you started spending your earnings. Be sure to tell what each variable in your equation represents.

y 5 48 2 3x;

x represents the time in days since I started spending my earnings, and y represents the amount of money I have left in dollars.

4. Use the coordinate plane to create a graph of your equation. Make sure to choose your bounds and intervals first. Label your graph clearly. Do not forget to name your graph. Extend your graph to show when your amount of earnings left would be 0.

Variable Quantity Lower Bound Upper Bound Interval

Time 0 30 2

Earnings left 0 60 4

y

2 4 6 8 10 x

0 12 14 16 18 20 2224 26 2830 4

8 12 16

0 20 24 28 32 36 40 44 48 52 56

60 Spending My Earnings

Time (days)

Earnings Left (dollars)

5. Determine the x- and y-intercepts of the graph. Explain what they mean in terms of the problem situation.

x-intercept: 16; y-intercept: 48

The x-intercept represents the number of days it will take to spend all the money.

The y-intercept represents the amount of money I started with.

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© 2011 Carnegie Learning

Grouping

Have students complete Questions 7 through 9 with a partner. Then share the response as a class.

Share Phase, Questions 7 and 8

What types of numbers should be considered when asked to determine the domain and range?

What does the domain mean with respect to the problem situation?

What does the range mean with respect to the problem situation?

What information in the problem situation helps you to determine the domain?

What information in the problem situation helps you to determine the range?

© 2011 Carnegie Learning

258 Chapter 4 Multiple Representations of Linear Functions

6. Is the slope of your line positive or negative? Does this make sense in terms of the problem situation?

Negative

Yes. The negative slope makes sense because the total amount of money is decreasing.

7. Consider your linear function without considering the problem situation. You can determine the domain of your linear function by using your

graph.

a. What is the domain of the function?

The domain is {all real numbers}. Because the graph is a straight non-vertical line, every input from the real numbers has exactly one output.

b. You can also determine the range by using your graph. What is the range of the function?

The range is {all real numbers}.

8. What do you think are the domain and range of any linear function of the form f(x) 5 mx 1 b? Explain your reasoning.

The domain and range of any linear function are all real numbers. On the graph of a straight non-vertical line, there is exactly one output for every input and the graph extends along the entire x-axis.

Recall that the set of all possible

input values of a function is the domain of the function and the set of all possible output values is the range of

the function.

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© 2011 Carnegie Learning

Is there a difference between the range of the graph of the equation and the range of the problem situation? Explain.

Notes

For some students, the equation y 5 48 2 3x may make more sense than y 5 2 3x 1 48 because it more closely models the context.

Make sure that they select the correct value for slope when dealing with this form of equation. If they decide to convert the equation to slope-intercept form, caution them of possible sign errors.

Problem 1 and Problem 2 lend themselves to having a discussion regarding mathematical models. A linear equation and its graph may model a context; the linear graph would be continuous and its domain and range would be all real numbers. A context may be more limited than its model; its graph may be a series of discrete points.

In that case, the domain and range would be more restricted than the domain and range of the model.

Distinguish between the domain and range of the algebraic model and the domain and range of the problem situation. Students should recognize and be able to answer questions related to the difference between the two concepts.

Share Phase, Question 9

Is there a difference between the domain of the graph of the equation and the domain of the problem situation?

Explain.

© 2011 Carnegie Learning

4.5 Developing the Graph of a Piecewise Function 259 9. Now consider your linear function again in terms of the problem situation.

a. What is the domain of the linear function in the problem situation? Explain your reasoning.

The domain is {all whole numbers from 0 to 16}. Because the input represents time in days, it does not make sense for the input to be negative or to be partial days. It also does not make sense to have whole numbers greater than 16 because the money is gone after the sixteenth day.

b. What is the range of the linear function in the problem situation?

Explain your reasoning.

The range is {all multiples of 3 between 0 and 48}. These are the output values for whole-number inputs from 0 to 16.

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© 2011 Carnegie Learning

Problem 2

The context in Problem 1 is modified to create a piecewise function. Students generate a table and graph from the modified context. The term piecewise function is defined.

They will create an algebraic representation of the piecewise function using appropriate notation, equations, and domains.

Grouping

Have students complete Questions 1 and 2 with a partner. Then share the responses as a class.

Share Phase, Questions 1 and 2

How is this context different from the context in Problem 1?

How is this context similar to the context in Problem 1?

Is the y-intercept the same in this situation as in Problem 1? Explain.

Is the slope the same in this situation as in Problem 1?

Explain.

Will the graph of this context be line? Explain.

Considering the values in the table, what do you think the graph will look like?

Is the x-intercept the same in this situation as in Problem 1? Explain.

Why is each piece of the graph a line segment?

© 2011 Carnegie Learning

260 Chapter 4 Multiple Representations of Linear Functions

Problem 2

Representing a Piecewise Function

Suppose that you do not spend the $48 by spending $3 each day. Instead, after 5 days of spending $3 each day, you do not spend anything for 5 days. You then spend $1.50 each day.

1. Complete the table to show the amounts of money you will have left for different numbers of days.

Time Since You Started Spending (days)

Amount of Earnings Left (dollars)

0 48

1 45

2 42

3 39

4 36

5 33

6 33

7 33

8 33

9 33

10 33

11 31.5

12 30

13 28.5

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© 2011 Carnegie Learning

Note

Students may examine the graph and want to include the values 5 and 10 in two pieces of the function. Guide them to use the context to determine to which piece of the function each value belongs.

Grouping

Ask a student to read the definition of piecewise function aloud. Discuss this information and complete Questions 3 through 5 as a class.

Discuss Phase,

Questions 3 through 5

Why is the domain of this context different from the domain of the context in Problem 1?

Where are the three distinct pieces of the function in your table?

How did you determine the domains of each piece of the function?

© 2011 Carnegie Learning

4.5 Developing the Graph of a Piecewise Function 261 2. Use the coordinate plane to create a graph from the table of values. Make sure to

choose your bounds and intervals first. Label your graph clearly. Extend your graph to show when your amount of earnings left would be 0.

Variable Quantity Lower Bound Upper Bound Interval

Time 0 45 3

Earnings left 0 60 4

y

3 6 9 12 15 x

0 18 21 24 27 30 3336 39 4245 4

8 12 16

0 20 24 28 32 36 40 44 48 52 56

60 Spending My Earnings

Time (days)

Earnings Left (dollars)

The graph that you created in Question 2 represents a piecewise function. A piecewise function is a function whose equation changes for different parts, or pieces, of the domain.

3. What is the domain of this function in the problem situation?

The domain is {all whole numbers from 0 to 32}.

4. How many pieces make up this function? (How many different equations are needed to describe this function?)

There are three pieces in the function.

5. What is the domain of each piece?

First piece: {0, 1, 2, 3, 4, 5}; Second piece: {6, 7, 8, 9, 10}; Third piece: {all whole numbers from 11 to 32}

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© 2011 Carnegie Learning

Grouping

Have students complete Questions 6 through 12 with a partner. Then share the responses as a class.

Share Phase,

Questions 6 through 9

What other method could you use to determine the equation for this piece of the function?

How could you use your graph to verify the accuracy of your equations?

Why does this function need brackets?

Why is it broken into three parts?

What does each of the inequalities represent?

Are the values 5 and 10 each used in two pieces of the function? Explain.

What expressions correspond with each domain?

Why are the domains listed in Question 9 different than the domains listed in Question 5?

© 2011 Carnegie Learning

262 Chapter 4 Multiple Representations of Linear Functions

6. Write an equation to represent the piece of the function from 0 days to 5 days. Show your work and explain how you determined your answer.

The problem statement gives the y-intercept, $48. Because $3 is being spent each day, the slope is 23. By using the slope-intercept form, the equation is y 5 23x 1 48.

7. Write an equation to represent the piece of the function from 6 days to 10 days. Show your work and explain your reasoning.

Because $0 is being spent each day, the slope of the line is 0. According to the table, the point (6, 33) is on the line. By using the point-slope form, the equation is y 2 33 5 0( x 2 6), or y 5 33.

8. Write an equation to represent the piece of the function from 11 days to 32 days.

Show your work and explain your reasoning.

According to the problem statement, $1.50 is being spent each day, so the slope is 21.5. According to the graph, the point (32, 0) is on the line. By using the point-slope form, the equation is y 2 0 5 21.5( x 2 32), or y 5 21.5x 1 48.

Recall, function notation can be used to write functions such that the dependent variable is replaced with the name of the function, such as f(x).

9. Complete the definition of your piecewise function, f( x).

For each piece of the domain, write the equation you wrote in Questions 6, 7, and 8.

23x 1 48 , 0  x  5

f( x) 5

5

33 , 5 , x  10

21.5x 1 48 , 10 , x  32

Could I use

g

(

x

) to name a

function?

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© 2011 Carnegie Learning

Share Phase,

Questions 10 through 12

How did you know which piece of the function to use to determine the x-intercept?

How did you know which piece of the function to use to determine the y-intercept?

How is determining the intercepts of a piecewise function different than determining the intercepts of a single function?

© 2011 Carnegie Learning

4.5 Developing the Graph of a Piecewise Function 263 10. Which piece should you use to determine the x-intercept? Which piece should you

use to determine the y-intercept?

The third piece should be used to determine the x-intercept.

The first piece should be used to determine the y-intercept.

11. Determine the intercepts of the graph. Show all your work.

x-intercept: y-intercept:

0 5 21.5x 1 48 y 5 2(3)0 1 48

248 5 21.5x y 5 0 1 48

x 5 32 y 5 48

12. When will you run out of money? How does this compare to the number of days it would take you to run out of money in Problem 1?

The money will be gone after 32 days. This is twice as long as it took in Problem 1.

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© 2011 Carnegie Learning

you felt you did not have enough information to place a point? How did you handle those cases?

Did you have to go back and revise your graph after completing it? Why and what revisions did you make?

Does your graph look different from your classmates’ graphs? Could more than one graph be correct? Explain.

Problem 3

Students graph a piecewise function from a context. The rigor in this problem is due to the multiple data points that need to be processed and the instances where students must determine reasonable rates for the activities described.

Grouping

Have students complete Question 1 with a partner. Then share the responses as a class.

Note

Struggling students may benefit by having this prose rewritten so that each fact is written on a separate line.

Share Phase, Question 1

Where is Kurt going? Does he make any stops along the way? Explain.

What is the time frame for Kurt’s trip to the community center?

How far is it to the community center?

How would drawing a diagram to represent Kurt’s path be helpful?

What do you think the graph of this situation will look like?

How many pieces will the graph have? Explain.

What order did you use when placing the points on your graph?

© 2011 Carnegie Learning

264 Chapter 4 Multiple Representations of Linear Functions

Problem 3

Modeling a Piecewise Function

1. Every Tuesday and Thursday, once Kurt gets home from school, he changes his clothes and goes to the community center, which is 1.9 miles from his house, to lift weights. He leaves his house at 3:25 pm and jogs at a steady rate for one mile to his friend Moe’s house, which is on the way to the community center. He stops and has a 10-minute break at Moe’s house, and then they walk at a consistent pace the rest of the way to the center. They arrive at the center at 4:10 pm.

a. Draw a graph on the grid provided that could show Kurt’s trip from home to the community center. Make sure to label the axes and show the intervals. Do not forget to name your graph.

x

4:00

1.2 1.6

4:10

3:20

0.8

3:40

3:00

0.4

3:50

3:10 3:30

2:50

y 1.8

1.0 1.4

0.6

0.2 0.0

Time of Day Kurt’s Trip to the Community Center

Distance from Home (miles) X

Y Z

The segment representing the first part of the trip should be steeper than the segment representing the third part of the trip. The y-axis may show Kurt’s distance from the community center.

b. Label the three pieces of the graph X, Y, and Z from left to right.

c. Order the pieces in terms of their slopes, from least to greatest.

Explain why you chose to order the pieces in that matter.

Y, Z, X

Y should be first because the slope is 0 when Kurt takes a 10-minute break at Moe’s house. Z should be next because that is when the boys are walking.

X should be last because the slope should be greatest when Kurt is jogging instead of walking.

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© 2011 Carnegie Learning

Grouping

Ask a student to read the context to Question 2 aloud.

Discuss the information and diagram as a class.

Have students complete Questions 2 and 3 with a partner. Then share the responses as a class.

Share Phase, Question 2

How is this problem the same as Problem 2?

How is this problem different from Problem 2?

How many miles is the hiking trip Lucy and her friends are taking? Explain.

What do you know about the time frame of the trip?

What do you think the graph of this context will look like?

How many pieces will the graph have? Describe them.

Will Lucy and her friends be travelling at the same rate during the entire trip?

Explain.

© 2011 Carnegie Learning

2. Lucy and her friends are hiking from their campsite to a waterfall. They leave their camp at 6:00 AM. They normally hike at a rate of 3 miles per hour, but on steeper parts of the hike, they slow down to 2 miles per hour. They have a one-hour picnic lunch by the waterfall and then reverse their path and hike back to the campsite.

The diagrams shown provide information about the trail that they will be hiking.

The diagrams are not drawn to scale.

Campsite Aerial View:

Side View:

6 miles

3 miles

Campsite

Picnic area Waterfall

Waterfall Picnic area

a. Draw a graph modeling Lucy’s hiking trip. Label the axes “Time of Day” and

“Distance from Campsite.” Make sure to show the intervals on the grid and do not forget to name your graph.

x

2 PM

6 8

10 AM

4

12 PM

8 AM

2

1 PM

9 AM 11 AM

7 AM

6 AM

y 9

5 7

3 A

B C

D

E

1 0

Time of Day Lucy’s Hiking Trip

Distance from Campsite (miles)

b. Label the pieces of the graph A, B, C, D, and E from left to right.

4.5 Developing the Graph of a Piecewise Function 265

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© 2011 Carnegie Learning 266 Chapter 4 Multiple Representations of Linear Functions

© 2011 Carnegie Learning

3. Examine your graph.

a. What pieces have negative slopes? Why are these slopes negative?

D and E each have a negative slope. The slopes are negative because when Lucy is traveling toward the camp, the distance to the camp is decreasing for each hour she travels.

b. Explain the relationship between the slopes of pieces B and D in terms of the problem situation.

B and D have the same numerical slope because they represent the same rate, 2 miles per hour. B has a positive slope because it is increasing because Lucy is moving away from the camp. D has a negative slope because it is decreasing because Lucy is moving toward the camp.

c. Explain the relationship between the slopes of pieces A and B in terms of the problem situation.

Both pieces have a positive slope because in both cases Lucy’s distance from camp is increasing. The slope of A is slightly greater than the slope of B because Lucy is moving at a faster rate, 3 miles per hour instead of 2 miles per hour.

d. Why is the slope of piece A greater than the slope of piece B in the graph, but in the diagram the reverse is true?

The slopes on the graph represent the rate of change in miles per hour. The diagram shows the terrain traveled. In the diagram, the steeper portion is B, but it is traveled at a slower rate, so its graph is flatter. In the diagram, the flatter portion is A, but it is traveled at a faster rate, so its graph is steeper.

e. Draw a vertical line through the graph to demonstrate its symmetry. Explain why there is symmetry in the graph.

A vertical line should be drawn through the point that is in the middle of lunchtime. There is symmetry because the same distance is traveled at the same rates both going to and returning from the waterfall.

Share Phase, Question 3

Does your graph match what you thought it would look like? Explain.

How did you decide what scale to use?

How did you decide where to place the points on your graph?

Did you have to go back and revise your graph after completing it? Why and what revisions did you make?

How did you decide whether each piece of the graph should be increasing or decreasing?

Can you visualize the story from your graph without referring to the context?

Is your graph more like the aerial view or the side view provided? Explain.

Misconception

Students may have the

misconception that the slope of segments D and E is negative because the hikers are going downhill. Although it is true that the hikers are going downhill, that is not the reason why the slope is negative. To clarify this concept, you may want to draw a different side view, one in which the hikers travel downhill with the return trip being uphill.

Then, have students see that this same hiking trail matches the graph they already have drawn.

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© 2011 Carnegie Learning

Problem 4

Students use a Calculator- Based Ranger (CBR) and a graphing calculator to act out piecewise functions.

Grouping

Have students complete Questions 1 through 4 with a partner. Then share the responses as a class.

© 2011 Carnegie Learning

4.5 Developing the Graph of a Piecewise Function 267

Problem 4

Acting Out a Piecewise Function

You will need a graphing calculator, a Calculator-Based Ranger (CBR), and a connector cable for this activity. You will also need a meter stick and masking tape to mark off distance measures.

Graphs of piecewise functions representing people walking, with time on the x-axis and distance on the y-axis, are shown after the step-by-step instructions. Your goal is to act out the graph by walking in the way that matches the graph. As you do this, your motion will be plotted alongside the graph to monitor your performance.

Step 1: Prepare the workspace.

Clear an area at least 1 meter wide and 4 meters long in front of a wall.

From the wall, measure the distances of 0.5, 1, 1.5, 2, 2.5, 3, 3.5, and 4 meters. Mark these distances on the floor using masking tape.

Step 2: Prepare the technology.

Connect the CBR to a graphing calculator.

Transfer the RANGER program from the CBR to the calculator. This only needs to be done the first time. It will then be stored in your calculator.

Press 2nd LINK ENTER.

Open the CBR and press the appropriate button on it for the type of calculator you are using. Your calculator screen will display RECEIVING and then DONE.

The CBR will flash a green light and beep.

Step 3: Access the RANGER program.

Press PRGM for program. Choose RANGER. Press ENTER .

Press ENTER to display the MAIN MENU.

Choose APPLICATIONS . Choose METERS .

Choose MATCH or DISTANCE MATCH .

Press ENTER . A graph will be displayed.

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© 2011 Carnegie Learning 268 Chapter 4 Multiple Representations of Linear Functions

© 2011 Carnegie Learning

Step 4: Act out the graph.

Examine the graph. Plan your path. Use the scale to gauge where to begin in relation to the wall. Will you walk toward or away from the wall? Will you walk fast or slow?

Hold the graphing calculator in one hand and the RANGER in the other hand.

The lid of the RANGER should be aimed toward the wall.

Press ENTER . Begin walking in a manner that matches the graph. Use the scale and floor markings as guides. You will hear a clicking sound and see a green light as your motion is plotted alongside the piecewise graph on the graphing calculator.

When the time is finished, examine your performance. What changes should you make?

Press ENTER to display the OPTIONS menu. Choose SAME MATCH .

Press ENTER and try the walk a second time.

Continue acting out walks by pressing ENTER and NEW MATCH.

When finished, press ENTER , choose MAIN MENU , and QUIT .

Time

x y = distance

Time

x y = distance

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© 2011 Carnegie Learning © 2011 Carnegie Learning

4.5 Developing the Graph of a Piecewise Function 269

x y = distance

Time

Time

x y = distance

x y

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© 2011 Carnegie Learning 270 Chapter 4 Multiple Representations of Linear Functions

© 2011 Carnegie Learning

1. How did you decide where to stand when beginning to act out a graph?

I had to look at the y-axis and use the scale to see how far from the wall I should stand. I used the floor markings to count out that distance.

2. How did you decide when to walk toward the wall and when to back up from the wall?

When the graph was increasing, I walked away from the wall because the distance from the wall was increasing.

When the graph was decreasing, I walked toward the wall because the distance from the wall was decreasing.

3. How did you act out a horizontal segment?

I stood still for a horizontal segment for the length of time designated on the x-axis.

4. How did you decide how fast to walk?

Using the scale and comparing my movement to previous parts of the graph were somewhat helpful. Comparing my movement with the graph while I was walking was more useful.

Be prepared to share your solutions and methods.

Did this activity help you make sense of the

graphs?

References

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